Hello Mary,

It will be helpful to begin with a few words on the rational and irrational numbers. The set of all real numbers can be represented as the union of 2 sets, the rational numbers and the irrational numbers. A **rational number is any number that can be expressed as a ratio of two integers**. In other words, a rational number is any number of the form p/q, where p and q are integers and q≠0. Basically, this just says that a rational number is an ordinary fraction. Of course, a rational number can also be expressed in decimal form. The **decimal form of a rational number either terminates or repeats.**

As you might expect, any real number which is not a rational number is called an **irrational number**. Specifically, this means an irrational number is a number that cannot be expressed in the form p/q, where p and q are integers. Alternately, an **irrational number can be defined as a number whose decimal form neither terminates, nor repeats.**

(After that lengthy digression, on to your questions.)

The "rule" you are given defines what is called a **piecewise function** (assuming it is indeed a function.) This means that different rules or formulas are used to give the output value (y-value) on different parts of the domain.

1. Every real number x is either rational or irrational (but it can't be both.) If x is rational, then y=1. If x is irrational, then y = 0. [For example, if x = 2/3 (a rational number), then y = 1. If x = π = 3.14159265.... (a famous example of an irrational number), then y = 0.] It is clear that **for each x-value, there corresponds exactly one y-value**. Therefore, this defines a **function**.

2. Recall that the domain is the set of all possible x-values ("inputs"). Since every real number is either a rational number or an irrational number, the function is defined for all real numbers. Therefore, the domain of this function is the set of **all real numbers**. If we wish, we can express this in interval notation as

**Domain = (-∞,∞)**.

3. The range is the set of all possible output values (y-values). Clearly, the only possible y-values are 0 or 1, so the **range is the set {0,1}**.

4. The y-intercept of any function can be found by setting x = 0. But 0 is a rational number, so the rule tells us that when x = 0, y = 1. Therefore, the y -intercept is **y = 1**. [Or, if required, we can express the y-intercept as the point **(0,1)**].

5. To find the x-intercept, set y=0. For what values of x does y = 0 for this function? The rule tells us that y = 0 if x is any irrational number. Consequently, **every irrational number is an x-intercept for this function**.

6. This part is a little tricky. Recall that a function is called even if f(-x) = f(x) for all x in its domain. We're not using "f(x)" notation in this function, but this is the same as saying that for every x in the domain, you obtain the same y-value from both x and -x. Geometrically, this says that the graph is symmetric with respect to the y-axis. Now to the main point:

Given any x, x is either rational or irrational.

Suppose x is rational. Then -x is also rational, so the y-value for both x and -x is 1.

Suppose x is irrational. Then -x is also irrational, so y = 0 for both x and -x.

(In either case, rational or irrational x, the y-value is the same for x and -x). Therefore, **the function is even.**

7. It's a bit difficult to describe the graph without showing it. Recall that if k is a constant, the graph of y=k is a horizontal line. If we only had the equation y = 1, we would have a horizontal line with y-intercept 1. Similarly, the function y = 0 is another horizontal line (corresponding to the x-axis.) The graph of our function can't consist of the two complete horizontal lines at y = 1 and y = 0. (If it did, the graph would not pass the vertical line test, so it would not represent a function.) However, our rule says that y equals either 0 or 1, but never at the same time (for the same value of x.) So the graph looks like the horizontal line y = 1 (but only over the rational values of x) and the horizontal line y = 0 (but only for the irrational numbers). We basically have the horizontal line y = 1 (but with all the points corresponding to irrational values of x removed), and the horizontal line y = 0 (missing all the points corresponding to rational number values of x.)

(As a result, this function is discontinuous for every real number!)

Hope that helps! Let me know if need any clarification.

William

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